Re: Integrate question

*To*: mathgroup at smc.vnet.net*Subject*: [mg77649] Re: [mg77600] Integrate question*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Thu, 14 Jun 2007 05:32:01 -0400 (EDT)*References*: <200706131144.HAA07285@smc.vnet.net>

On 13 Jun 2007, at 20:44, dimitris wrote: > This question aose after replying in the forum for another CAS. > In fact, it is not a question on "how I do something" but rather > asking about some information about "why". > > In view of my reply I realized one fundamental difference > between Integrate and the function Int from another CAS. > > int( diff(y(x),x) , x=a..b); > > b > / > | d > | -- y(x) dx > | dx > / > a > > In[11]:= > Integrate[Derivative[1][f][x], {x, a, b}] > Out[11]= > -f[a]+f[b] > > I guess all we are familiar with the first fundamental theorem of > calculus. > It holds (http://mathworld.wolfram.com/ > FundamentalTheoremsofCalculus.html) > if the function f[x] is continuous on the closed interval [a,b] (or at > least piecewise > continuous in Newton-Leibniz form with limits left and right of the > finite discontinuities). > I am with Mathematica default design. For example > > In[12]:= > Integrate[Derivative[1][f][x]*Derivative[2][f][x], {x, a, b}] > > Out[12]= > (1/2)*(-Derivative[1][f][a]^2 + Derivative[1][f][b]^2) > > In[13]:= > Integrate[Derivative[1][f][x]*g[x] + f[x]*Derivative[1][g][x], {x, a, > b}] > > Out[13]= > (-f[a])*g[a] + f[b]*g[b] > > In[14]:= > Integrate[Sin[f[x]]*Derivative[1][f][x], {x, a, b}] > > Out[14]= > Cos[f[a]] - Cos[f[b]] > > I am neither software enginner, nor pure mathematician but this > fundmental difference impressed me a lot! I am familiar > with the issue of generic complex values in mathematica > but here Mathematica "assumes" that typing Integrate[f'[x],{x,a,b}] > f[x] is continuous in [a,b]? > > I will appreciate any comments on this issue. > > Greeting from the sunny Athens, > > Dimitris > > Given a general functions f, without any special properties Mathematica will assume that it is in fact an analytic function; which is why you get: Series[f[x], {x, 0, 3}] SeriesData[x, 0, {f[0], Derivative[1][f][0], Derivative[2][f][0]/2, Derivative[3][f][0]/6}, 0, 4, 1] etc. So in particular, an undefined function is always infinitely differentiable everywhere and hence continuous. Series has an option, which can be used to remove this assumption: Series[f[x]*Sin[x], {x, 0, 3}, Analytic -> False] f[x]*SeriesData[x, 0, {1, 0, -1/6}, 1, 4, 1] In other situations, e.g with Integrate, or D, there is no analogous option but then I can't see any situation in which it could be useful in the context of computer algebra. Can anyone? Andrzej Kozlowski although, to tell the truth, I can't think of any serious use for this in the context of "computer algebra" (can anyone?) Andrzej Kozlowski

**References**:**Integrate question***From:*dimitris <dimmechan@yahoo.com>